3.445 \(\int \frac{\left (c+d x^2\right )^3}{x^{3/2} \left (a+b x^2\right )} \, dx\)

Optimal. Leaf size=284 \[ -\frac{(b c-a d)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{5/4} b^{11/4}}+\frac{(b c-a d)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{5/4} b^{11/4}}+\frac{(b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{5/4} b^{11/4}}-\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{5/4} b^{11/4}}+\frac{2 d^2 x^{3/2} (3 b c-a d)}{3 b^2}-\frac{2 c^3}{a \sqrt{x}}+\frac{2 d^3 x^{7/2}}{7 b} \]

[Out]

(-2*c^3)/(a*Sqrt[x]) + (2*d^2*(3*b*c - a*d)*x^(3/2))/(3*b^2) + (2*d^3*x^(7/2))/(
7*b) + ((b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^
(5/4)*b^(11/4)) - ((b*c - a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/
(Sqrt[2]*a^(5/4)*b^(11/4)) - ((b*c - a*d)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4
)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(5/4)*b^(11/4)) + ((b*c - a*d)^3*Log[Sqrt[a
] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(5/4)*b^(11/4))

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Rubi [A]  time = 0.5904, antiderivative size = 284, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{(b c-a d)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{5/4} b^{11/4}}+\frac{(b c-a d)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{5/4} b^{11/4}}+\frac{(b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{5/4} b^{11/4}}-\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{5/4} b^{11/4}}+\frac{2 d^2 x^{3/2} (3 b c-a d)}{3 b^2}-\frac{2 c^3}{a \sqrt{x}}+\frac{2 d^3 x^{7/2}}{7 b} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)^3/(x^(3/2)*(a + b*x^2)),x]

[Out]

(-2*c^3)/(a*Sqrt[x]) + (2*d^2*(3*b*c - a*d)*x^(3/2))/(3*b^2) + (2*d^3*x^(7/2))/(
7*b) + ((b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^
(5/4)*b^(11/4)) - ((b*c - a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/
(Sqrt[2]*a^(5/4)*b^(11/4)) - ((b*c - a*d)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4
)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(5/4)*b^(11/4)) + ((b*c - a*d)^3*Log[Sqrt[a
] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(5/4)*b^(11/4))

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Rubi in Sympy [A]  time = 110.388, size = 265, normalized size = 0.93 \[ \frac{2 d^{3} x^{\frac{7}{2}}}{7 b} - \frac{2 d^{2} x^{\frac{3}{2}} \left (a d - 3 b c\right )}{3 b^{2}} - \frac{2 c^{3}}{a \sqrt{x}} + \frac{\sqrt{2} \left (a d - b c\right )^{3} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 a^{\frac{5}{4}} b^{\frac{11}{4}}} - \frac{\sqrt{2} \left (a d - b c\right )^{3} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 a^{\frac{5}{4}} b^{\frac{11}{4}}} - \frac{\sqrt{2} \left (a d - b c\right )^{3} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{5}{4}} b^{\frac{11}{4}}} + \frac{\sqrt{2} \left (a d - b c\right )^{3} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{5}{4}} b^{\frac{11}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)**3/x**(3/2)/(b*x**2+a),x)

[Out]

2*d**3*x**(7/2)/(7*b) - 2*d**2*x**(3/2)*(a*d - 3*b*c)/(3*b**2) - 2*c**3/(a*sqrt(
x)) + sqrt(2)*(a*d - b*c)**3*log(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) +
sqrt(b)*x)/(4*a**(5/4)*b**(11/4)) - sqrt(2)*(a*d - b*c)**3*log(sqrt(2)*a**(1/4)*
b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(4*a**(5/4)*b**(11/4)) - sqrt(2)*(a*d -
b*c)**3*atan(1 - sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(2*a**(5/4)*b**(11/4)) + sqr
t(2)*(a*d - b*c)**3*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(2*a**(5/4)*b**(
11/4))

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Mathematica [A]  time = 0.186454, size = 283, normalized size = 1. \[ \frac{(a d-b c)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{5/4} b^{11/4}}+\frac{(b c-a d)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{5/4} b^{11/4}}+\frac{(b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{5/4} b^{11/4}}+\frac{(a d-b c)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{5/4} b^{11/4}}+\frac{2 d^2 x^{3/2} (3 b c-a d)}{3 b^2}-\frac{2 c^3}{a \sqrt{x}}+\frac{2 d^3 x^{7/2}}{7 b} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)^3/(x^(3/2)*(a + b*x^2)),x]

[Out]

(-2*c^3)/(a*Sqrt[x]) + (2*d^2*(3*b*c - a*d)*x^(3/2))/(3*b^2) + (2*d^3*x^(7/2))/(
7*b) + ((b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^
(5/4)*b^(11/4)) + ((-(b*c) + a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)
])/(Sqrt[2]*a^(5/4)*b^(11/4)) + ((-(b*c) + a*d)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*
b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(5/4)*b^(11/4)) + ((b*c - a*d)^3*Log[
Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(5/4)*b^(11
/4))

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Maple [B]  time = 0.018, size = 622, normalized size = 2.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)^3/x^(3/2)/(b*x^2+a),x)

[Out]

2/7*d^3*x^(7/2)/b-2/3*d^3/b^2*x^(3/2)*a+2*d^2/b*x^(3/2)*c+1/2*a^2/b^3/(a/b)^(1/4
)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*d^3-3/2*a/b^2/(a/b)^(1/4)*2^(1/2
)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*c*d^2+3/2/b/(a/b)^(1/4)*2^(1/2)*arctan(2
^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*c^2*d-1/2/a/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/
b)^(1/4)*x^(1/2)-1)*c^3+1/4*a^2/b^3/(a/b)^(1/4)*2^(1/2)*ln((x-(a/b)^(1/4)*x^(1/2
)*2^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))*d^3-3/4*a/b^
2/(a/b)^(1/4)*2^(1/2)*ln((x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1
/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))*c*d^2+3/4/b/(a/b)^(1/4)*2^(1/2)*ln((x-(a/b)^(1
/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))*c^
2*d-1/4/a/(a/b)^(1/4)*2^(1/2)*ln((x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x+
(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))*c^3+1/2*a^2/b^3/(a/b)^(1/4)*2^(1/2)*ar
ctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*d^3-3/2*a/b^2/(a/b)^(1/4)*2^(1/2)*arctan(2^(
1/2)/(a/b)^(1/4)*x^(1/2)+1)*c*d^2+3/2/b/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)
^(1/4)*x^(1/2)+1)*c^2*d-1/2/a/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(
1/2)+1)*c^3-2*c^3/a/x^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^3/((b*x^2 + a)*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.280197, size = 2843, normalized size = 10.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^3/((b*x^2 + a)*x^(3/2)),x, algorithm="fricas")

[Out]

1/42*(12*a*b*d^3*x^4 - 84*b^2*c^3 + 84*a*b^2*sqrt(x)*(-(b^12*c^12 - 12*a*b^11*c^
11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^
5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8
- 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^
5*b^11))^(1/4)*arctan(-a^4*b^8*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^1
0*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^
6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9
+ 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^5*b^11))^(3/4)/((b^9*c
^9 - 9*a*b^8*c^8*d + 36*a^2*b^7*c^7*d^2 - 84*a^3*b^6*c^6*d^3 + 126*a^4*b^5*c^5*d
^4 - 126*a^5*b^4*c^4*d^5 + 84*a^6*b^3*c^3*d^6 - 36*a^7*b^2*c^2*d^7 + 9*a^8*b*c*d
^8 - a^9*d^9)*sqrt(x) - sqrt((b^18*c^18 - 18*a*b^17*c^17*d + 153*a^2*b^16*c^16*d
^2 - 816*a^3*b^15*c^15*d^3 + 3060*a^4*b^14*c^14*d^4 - 8568*a^5*b^13*c^13*d^5 + 1
8564*a^6*b^12*c^12*d^6 - 31824*a^7*b^11*c^11*d^7 + 43758*a^8*b^10*c^10*d^8 - 486
20*a^9*b^9*c^9*d^9 + 43758*a^10*b^8*c^8*d^10 - 31824*a^11*b^7*c^7*d^11 + 18564*a
^12*b^6*c^6*d^12 - 8568*a^13*b^5*c^5*d^13 + 3060*a^14*b^4*c^4*d^14 - 816*a^15*b^
3*c^3*d^15 + 153*a^16*b^2*c^2*d^16 - 18*a^17*b*c*d^17 + a^18*d^18)*x - (a^3*b^17
*c^12 - 12*a^4*b^16*c^11*d + 66*a^5*b^15*c^10*d^2 - 220*a^6*b^14*c^9*d^3 + 495*a
^7*b^13*c^8*d^4 - 792*a^8*b^12*c^7*d^5 + 924*a^9*b^11*c^6*d^6 - 792*a^10*b^10*c^
5*d^7 + 495*a^11*b^9*c^4*d^8 - 220*a^12*b^8*c^3*d^9 + 66*a^13*b^7*c^2*d^10 - 12*
a^14*b^6*c*d^11 + a^15*b^5*d^12)*sqrt(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^
10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 +
924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^
3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^5*b^11))))) + 21
*a*b^2*sqrt(x)*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*
b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 -
792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^
2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^5*b^11))^(1/4)*log(a^4*b^8*(-(b^12*c^1
2 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*
c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*
a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11
+ a^12*d^12)/(a^5*b^11))^(3/4) - (b^9*c^9 - 9*a*b^8*c^8*d + 36*a^2*b^7*c^7*d^2 -
84*a^3*b^6*c^6*d^3 + 126*a^4*b^5*c^5*d^4 - 126*a^5*b^4*c^4*d^5 + 84*a^6*b^3*c^3
*d^6 - 36*a^7*b^2*c^2*d^7 + 9*a^8*b*c*d^8 - a^9*d^9)*sqrt(x)) - 21*a*b^2*sqrt(x)
*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 +
495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^
5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^
11*b*c*d^11 + a^12*d^12)/(a^5*b^11))^(1/4)*log(-a^4*b^8*(-(b^12*c^12 - 12*a*b^11
*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792
*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d
^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/
(a^5*b^11))^(3/4) - (b^9*c^9 - 9*a*b^8*c^8*d + 36*a^2*b^7*c^7*d^2 - 84*a^3*b^6*c
^6*d^3 + 126*a^4*b^5*c^5*d^4 - 126*a^5*b^4*c^4*d^5 + 84*a^6*b^3*c^3*d^6 - 36*a^7
*b^2*c^2*d^7 + 9*a^8*b*c*d^8 - a^9*d^9)*sqrt(x)) + 28*(3*a*b*c*d^2 - a^2*d^3)*x^
2)/(a*b^2*sqrt(x))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**2+c)**3/x**(3/2)/(b*x**2+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.286885, size = 624, normalized size = 2.2 \[ -\frac{2 \, c^{3}}{a \sqrt{x}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{3}{4}} a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, a^{2} b^{5}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{3}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, a^{2} b^{5}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{3}{4}} a^{3} d^{3}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, a^{2} b^{5}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{3}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{3}{4}} a^{3} d^{3}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, a^{2} b^{5}} + \frac{2 \,{\left (3 \, b^{6} d^{3} x^{\frac{7}{2}} + 21 \, b^{6} c d^{2} x^{\frac{3}{2}} - 7 \, a b^{5} d^{3} x^{\frac{3}{2}}\right )}}{21 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^3/((b*x^2 + a)*x^(3/2)),x, algorithm="giac")

[Out]

-2*c^3/(a*sqrt(x)) - 1/2*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*
c^2*d + 3*(a*b^3)^(3/4)*a^2*b*c*d^2 - (a*b^3)^(3/4)*a^3*d^3)*arctan(1/2*sqrt(2)*
(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a^2*b^5) - 1/2*sqrt(2)*((a*b^3)^
(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/4)*a^2*b*c*d^2 - (a*b
^3)^(3/4)*a^3*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(
1/4))/(a^2*b^5) + 1/4*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2
*d + 3*(a*b^3)^(3/4)*a^2*b*c*d^2 - (a*b^3)^(3/4)*a^3*d^3)*ln(sqrt(2)*sqrt(x)*(a/
b)^(1/4) + x + sqrt(a/b))/(a^2*b^5) - 1/4*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*
b^3)^(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/4)*a^2*b*c*d^2 - (a*b^3)^(3/4)*a^3*d^3)*ln
(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^2*b^5) + 2/21*(3*b^6*d^3*x^(7/
2) + 21*b^6*c*d^2*x^(3/2) - 7*a*b^5*d^3*x^(3/2))/b^7